Indefinite Morse 2-functions; broken fibrations and generalizations

Morse 同调中链级模型的 Chas-Sullivan 乘积研究

A chain-level model for Chas-Sullivan products in Morse homology with differential graded coefficients

本文提出了一种基于 Morse 理论的链级模型，用于研究自由循环空间的 Chas-Sullivan 乘积，并构建了微分格版本的 Künneth 公式和 Pontryagin-Thom 构造。

Diffeomorphisms preserving Morse-Bott functions

Let $f:M\to\mathbb{R}$ be a Morse-Bott function on a closed manifold $M$, so the set $Σ_f$ of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by $\mathcal{S}(f) = \{h \in \mathcal{D}(M) \mid f\circ h=h \}$ the group of diffeomorphisms of $M$ preserving $f$ and let $\mathcal{D}(Σ_f)$ be the group of diffeomorphisms of $Σ_f$. We prove that the "restriction to $Σ_f$" map $ρ:\mathcal{S}(f) \to \mathcal{D}(Σ_f)$, $ρ(h) = h|_{Σ_f}$, is a locally trivial fibration over its image $ρ(\mathcal{S}(f))$.

链环的Morse-Novikov理论

Morse-Novikov theory for links

研究S^3中链环补的Thurston多面体的纤维面，利用非交换Novikov同调证明二桥链环的纤维化判据，并计算多数低交叉数素链环的Morse-Novikov数。

双曲6流形上的完美圆值Morse函数

Perfect circle-valued Morse functions on hyperbolic 6-manifolds

本文首次构造了具有完美圆值Morse函数的双曲6流形，并由此得到双曲群中一个类型为F2而非F3的子群新例子。

Modification rule of monodromies in R_2-move

An R_2-move is a homotopy of wrinkled fibrations which deforms images of indefinite fold singularities like Reidemeister move of type II. Variants of this move are contained in several important deformations of wrinkled fibrations, flip and slip for example. In this paper, we first investigate how monodromies are changed by this move. For a given fibration and its vanishing cycles, we then give an algorithm to obtain vanishing cycles in one reference fiber of a fibration, which is obtained by applying flip and slip to the original fibration, in terms of mapping class groups. As an application …

论四维Dehn扭转与Milnor纤维化

On four-dimensional Dehn twists and Milnor fibrations

该研究借助磁单极Floer同调进展计算族Seiberg--Witten不变量，证明极小椭圆奇点链不定辛填充的边界Dehn扭转无限阶非平凡，揭示了四维空间中三类全新的拓扑与辛几何现象。

Counting special Lagrangian fibrations in twistor families of K3 surfaces

The number of closed billiard trajectories in a rational-angled polygon grows quadratically in the length. This paper gives an analogue on K3 surfaces, by considering special Lagrangian tori. The analogue of the angle of a billiard trajectory is a point on a twistor sphere, and the number of directions admitting a special Lagrangian torus fibration with volume bounded by $ V $ grows like $ V^{20} $ with a power-saving term. Bergeron--Matheus have explicitly estimated the exponent of the error term as $ {20-\frac{4}{697633} }$. The counting result on K3 surfaces is deduced from a count of primi…

Circle-valued Morse theory for frame spun knots and surface-links

Let N be a closed oriented k-dimensional submanifold of the (k+2)-dimensional sphere; denote its complement by C(N). Denote by x the 1-dimensional cohomology class in C(N), dual to N. The Morse-Novikov number of C(N) is by definition the minimal possible number of critical points of a regular Morse map f from C(N) to a circle, such that f belongs to x. In the first part of this paper we study the case when N is the twist frame spun knot associated to an m-knot K. We obtain a formula which relates the Morse-Novikov numbers of N and K and generalizes the classical results of D. Roseman and E.C. …

The Picard-Lefschetz theory of complexified Morse functions

Given a closed manifold N and a self-indexing Morse function f: N --> R with up to four distinct Morse indices, we construct a symplectic Lefschetz fibration pi: E --> C which models the complexification of f on the disk cotangent bundle, f_C : D(T*N) --> C, when f is real analytic. By construction, pi: E --> C comes with an explicit regular fiber M and vanishing spheres V_1,...,V_m in M, one for each critical point of f. Our main result is that (E,pi) is a good model for the complexification (D(T*N),f_C) in the sense that N embeds in E as an exact Lagrangian submanifold, and in addition, pi|N…

Complexity functions on 1-dimensional cohomology

For a smooth, closed $n$-manifold $M$, we define an upper semi-continuous integer-valued complexity function on $H^1(M;{\mathbb R})$ using Morse theory. This measures how far an integral class is from being a fiber of a fibration. The fact complexity minimisers are open generalises Tischler's result on the openness of classes dual to fibrations. We then use this to define a complexity function on 1-dimensional cohomology of a finitely presented group, which is constant on open rays from the origin and vanishes precisely on the geometric invariant due to Bieri, Neumann and Strebel.

Homotopy types of stabilizers and orbits of Morse functions on surfaces

Let $M$ be a smooth compact surface, orientable or not, with boundary or without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$ the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule $h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in C^{\infty}(M,P)$. Let $f:M \to P$ be a Morse function and $O(f)$ be the orbit of $f$ under this action. We prove that $π_k O(f)=π_k M$ for $k\geq 3$, and $π_2 O(f)=0$ except for few cases. In particular, $O(f)$ is aspherical, provided so is $M$. Moreover, $π_1 O(f)$ is an extension of a finitely generated …

Topology of spaces of smooth functions and gradient-like flows with prescribed singularities on surfaces

By a gradient-like flow on a closed orientable surface $M$, we mean a closed 1-form $β$ defined on $M$ punctured at a finite set of points (sources and sinks of $β$) such that there exists a Morse function $f$ on $M$, called an energy function of $β$, whose critical points coincide with equilibria of $β$, and the pair $(f,β)$ has a canonical form near each critical point of $f$. Let $\mathcal{B}=\mathcal{B}(β_0)$ be the space of all gradient-like flows on $M$ having the same types of local singularities as a flow $β_0$, and $\mathcal{F}=\mathcal{F}(f_0)$ the space of all Morse functions on $M$…